Weighted ${\ell }_q$ approximation problems on the ball and on the sphere

Abstract

Let $L_{q,\mu },1\le q<\infty ,\mu \ge 0,$ denote the weighted $L_q$ space with the classical Jacobi weight $w_{\mu }$ on the ball ${\mathbb{B}}^d$. We consider the weighted least ${\ell }_q$ approximation problem for a given $L_{q,\mu }$-Marcinkiewicz-Zygmund family on ${\mathbb{B}}^d$. We obtain the weighted least ${\ell }_q$ approximation errors for the weighted Sobolev space $W_{q,\mu }^r$, $r>(d+2\mu )/q$, which are order optimal. We also discuss the least squares quadrature induced by an $L_{2,\mu }$-Marcinkiewicz-Zygmund family, and get the quadrature errors for $W_{2,\mu }^r$, $r>(d+2\mu )/2$, which are also order optimal. Meanwhile, we give the corresponding the weighted least ${\ell }_q$ approximation theorem and the least squares quadrature errors on the sphere.

17 pages